# Critical behaviour of the XY -rotors model on regular and small world networks

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* Corresponding author 1 CPT - E5 Physique statistique et systèmes complexes CPT - Centre de Physique Théorique - UMR 7332 2 CPT - Centre de Physique Théorique - UMR 7332

Abstract : We study the $XY$-rotors model on small networks whose number of links scales with the system size $N {links}\sim N^{\gamma}$, where $1\le\gamma\le2$. We first focus on regular one dimensional rings in the microcanonical ensemble. For $\gamma1.5$, the system equilibrium properties are found to be identical to the mean field, which displays a second order phase transition at a critical energy density $\varepsilon=E-N,~\varepsilon {c}=0.75$. Moreover for $\gamma {c}\simeq1.5$ we find that a non trivial state emerges, characterized by an infinite susceptibility. We then consider small world networks, using the Watts-Strogatz mechanism on the regular networks parametrized by $\gamma$. We first analyze the topology and find that the small world regime appears for rewiring probabilities which scale as $p {SW}\propto1-N^{\gamma}$. Then considering the $XY$-rotors model on these networks, we find that a second order phase transition occurs at a critical energy $\varepsilon {c}$ which logarithmically depends on the topological parameters $p$ and $\gamma$. We also define a critical probability $p {MF}$, corresponding to the probability beyond which the mean field is quantitatively recovered, and we analyze its dependence on $\gamma$.

Keywords : phase transition networks XY model

Autor: Sarah De Nigris - Xavier Leoncini -

Fuente: https://hal.archives-ouvertes.fr/

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